Carlet’s Kasami cyclic-additive conjecture
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For a field K of characteristic two with |K|=2ⁿ and gcd(k,n)=1, set dₖ=4ᵏ−2ᵏ+1 and Δ={(b+1)^dₖ+b^dₖ+1:b∈K}. Does v₁x+v₂y+(v₁+v₂)z=0 have exactly 2^(2n−3) solutions in Δ³ for all distinct nonzero v₁,v₂?
Why it matters and what counts as progress
Why it matters: The uniform Kasami count extends selected residue-class results to the full finite-field parameter range.
What counts as progress: Prove the Kasami cyclic-additive count for all admissible parameters or exhibit a parameter pair with a different count.
How the problem developed
Claude Carlet Introduced the cyclic-additive difference-set question for Kasami monomials as a condition related to componentwise Walsh uniformity.
- 2019 · Open problem [2]
NSUCRYPTO problem authors Presented the cyclic-additive question in the Kasami setting as a research problem.
- 6 Oct 2026 · Reported theorem [2]
Nagy, McNeil, and Vajda Proved the cyclic-additive conjecture for all admissible parameters, extending earlier congruence-class cases and bounded computations.
AI contributions 2
Carlet's cyclic-additive conjecture for the Kasami monomials
Let K be a field of characteristic two with |K|=2ⁿ, gcd(k,n)=1 and dₖ=4ᵏ−2ᵏ+1. If Δ is the image of b↦(b+1)^dₖ+b^dₖ+1, then for distinct nonzero v₁,v₂∈K exactly 2^(2n−3) triples (x,y,z)∈Δ³ satisfy v₁x+v₂y+(v₁+v₂)z=0.
On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for k mod n ∈ {1, 2, n−2, n−1}, and exhaustive verification for n ≤ 13
For the Kasami function F(x)=x^(4^k−2^k+1) over F_(2^n), with gcd(k,n)=1, Nagy and Vajda prove the conjectured ordered triple count 2^(2n−3) for all distinct nonzero coefficients when k mod n is 1, 2, n−2 or n−1. They also check admissible n≤13 by exhaustive computation.
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AI-assisted proof resolves Carlet’s Kasami cyclic-additive conjecture